Tarskian Semantics

Origin. Tarski (1933). Model-theoretic truth definition. Recursive satisfaction. Convention T. Foundation for formal semantics.

Models. Structures interpret language. Satisfaction recursively defined. Truth = satisfaction by all assignments. Compositionality.

Formalism.

Structure: M = (D, I) D: domain (non-empty) I: interpretation of symbols

Variable assignment: σ : Var → D Maps variables to domain elements.

Term interpretation: ⟦x⟧^M_σ = σ(x) ⟦c⟧^M_σ = I(c) ⟦f(t₁,...,tₙ)⟧^M_σ = I(f)(⟦t₁⟧,...,⟦tₙ⟧)

Satisfaction (base): M, σ ⊨ P(t₁,...,tₙ) iff (⟦t₁⟧,...,⟦tₙ⟧) ∈ I(P) M, σ ⊨ t₁ = t₂ iff ⟦t₁⟧ = ⟦t₂⟧

Satisfaction (connectives): M, σ ⊨ ¬φ iff M, σ ⊭ φ M, σ ⊨ φ ∧ ψ iff M, σ ⊨ φ and M, σ ⊨ ψ M, σ ⊨ φ ∨ ψ iff M, σ ⊨ φ or M, σ ⊨ ψ M, σ ⊨ φ → ψ iff M, σ ⊭ φ or M, σ ⊨ ψ

Satisfaction (quantifiers): M, σ ⊨ ∀x.φ iff M, σ[x↦d] ⊨ φ for all d ∈ D M, σ ⊨ ∃x.φ iff M, σ[x↦d] ⊨ φ for some d ∈ D

Truth: M ⊨ φ iff M, σ ⊨ φ for all σ. Sentence true if satisfied under all assignments.

Convention T: "Snow is white" is true iff snow is white. Material adequacy condition.

Symbols.

Metatheory. Completeness: ⊢ iff ⊨. Soundness. Truth undefinable in own language (Tarski). Object/metalanguage distinction.

Applies to. Model theory. Formal semantics. Philosophy of language. Logic foundations.

Limitations. Object/meta distinction. Self-reference restricted. Intensionality challenges. Natural language gaps.

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